Exact solution of the Jaynes-Cummings model with cavity damping

Abstract

Operating in Laplace language and making use of a representation based on photon-number states, we find the exact solution for the density operator that belongs to the Jaynes-Cummings model with cavity damping. The detuning parameter is set equal to zero and the optical resonator does not contain any thermal photons. It is shown that the master equation for the density operator can be replaced by two algebraic recursion relations for vectors of dimension 2 and 4. These vectors are built up from suitably chosen matrix elements of the density operator. By performing an iterative procedure, the exact solution for each matrix element is found in the form of an infinite series. We demonstrate that all series are convergent and discuss how they can be truncated when carrying out numerical work. With the help of techniques from function theory, it is proved that our solutions respect the following conditions on the density operator: conservation of trace, Hermiticity, convergence to the initial state for small times, and convergence to the ground state for large times. We compute some matrix elements of the density operator for the case of weak damping and find that their analytic structure becomes much simpler. Finally, it is shown that the exact atomic density matrix converges to the state of maximum von Neumann entropy if the time, the square of the initial electromagnetic energy density, and the inverse of the cavity-damping parameter tend to infinity equally fast. The initial condition for the atom can be chosen freely, whereas the field may start from either a coherent or a photon-number state.

Closed-form solution of Lindblad master equations without gain

“…Efforts have been made to tackle this problem and, while there exist analytical steady state solutions to some problems [9], there are not many solutions to the eigenvalue problem of specific systems. The dissipative version of the Harmonic oscillator, up to two spins and the Jaynes-Cummings model are the only open systems for which exact solutions of the eigenvalue problem are known [10][11][12][13][14][15]. The solutions to the Jaynes-Cummings model that can be found in the literature [10,14] are examples that show how intricate the calculation of the eigensystem of the Liouville operator can be.…”

Quantum Information Features Attendant to Atomic Spontaneous Decay

“…Such a generalization is of considerable interest because of its relevance to the study of the nonlinear coupling between a two-level atom and the radiation field [6,7,8,9,10]. Much work has been concentrated to the theoretical study of the dissipative JC model [11,12,13] by considering cavities losses and atomic decay. Also, there are some experiments that have shown decoherence of the superposition of states due to the interaction of the system with the environment [14].…”

“…Also, there are some experiments that have shown decoherence of the superposition of states due to the interaction of the system with the environment [14]. After introducing of the JC model, attention has been focused on some dissipative variants of this model in the last three decades [15,16,17,18,19]. The experiments with highly excited Rydberg atoms allowed some of the predictions of the extended versions of the JC model to be observed.…”

Jaynes-Cummings-model with damping at resonance

“…(6) Because of γ L γ A many authors (e.g. [12][13][14]) neglect the damping of the atom. However, we will show, that this can straightforwardly be included in the first order as long as the sum γ A + γ P is not significantly larger than γ L .…”

“…We are not only interested in the evolution of the density matrix as given by equation (5) or of equivalent quasi probabilities [12][13][14] but also in the dynamics of observables.…”